President Obama summoned House Speaker John A. Boehner to the White House on Thursday, but their talks over the looming “fiscal cliff” failed to break the impasse over taxes and clear the way for a deal to head off painful austerity measures set to hit in January.

After the 50-minute session in the Oval Office, aides to both men described the meeting as a frank exchange and said the lines of communication remained open. But Boehner prepared to return to Ohio for the weekend, with no further negotiations on his schedule.

The late-afternoon invitation from the president surprised senior Republicans. But it did nothing to dispel the sense that hope is all but lost for a far-reaching agreement to raise taxes, cut spending and avert the year-end cliff.

Obama’s top economic adviser, Gene Sperling, confirmed the gloomy outlook in a closed-door lunch meeting Thursday with Senate Democrats, according to those present.

Senior Senate Republicans, meanwhile, were at work on a fallback plan that would not significantly restrain the national debt but would at least avert widespread economic damage by canceling tax increases scheduled to take effect next year for the vast majority of Americans. That strategy calls for Republicans to capitulate to Obama’s demand to let tax rates rise on wage and salary income for the wealthiest 2 percent of taxpayers.

But the approach would also seek to thwart the Democrats by trying to block other steps that would increase taxes paid by wealthy taxpayers, including higher rates on investment income and limits on the value of itemized deductions. This strategy would produce only about $440 billion in new taxes and give the Democrats even less revenue than Republicans had previously put on the table. In his initial offer earlier this month, Boehner had said he could support $800 billion in new tax revenue.

With a relatively low price in new taxes, the strategy, if successful, would represent a tactical victory for Republicans and shift the political burden onto Democrats to make greater concessions on federal spending.

Senate Minority Leader Mitch McConnell (R-Ky.) has floated the strategy among Republicans in both chambers, according to lawmakers and senior GOP aides. It has been rejected by Boehner and other House leaders, aides said.

Don Stewart, a spokesman for McConnell, denied that he was floating the idea. “The leader has held numerous discussions to hear input from his members, and many ideas have been raised,” he said.

But top GOP policy aides continued to refine the plan Thursday as a growing number of Senate Republicans clamored for action to prevent the fiscal crisis — for which, polls show, Republicans would be blamed.

“It’s bizarre. It’s truly bizarre. There’s no sense of urgency,” said retiring Sen. Olympia J. Snowe (R-Maine). “We’re lurching from deadline to deadline now. We do not want to create this as the new norm.”

The divide between Republicans in the House and Senate has become increasingly apparent in recent days. While House leaders have roundly rejected any strategy that would let tax rates rise, the ranks of Senate Republicans who are willing to accept a small tax hike and continue the fight over spending early next year continue to grow.

On Thursday, Sen. Richard Burr (R-N.C.), a close friend of Boehner and a reliable conservative, added his voice to the chorus.

“We’re for extending the middle-class tax rates. We can debate the upper-end tax rates when we do tax reform. But if we’ve got to do this to get the president focused on reform of taxes, preservation of entitlements — Medicare, Social Security and Medicaid — I’m all for it,” said Burr, echoing comments from GOP colleagues, including John Cornyn (Tex.), who is ascending to the No. 2 slot in leadership.

Just eight House Republicans — mostly moderates — have publicly expressed support for something akin to the McConnell strategy, according to an internal tally maintained by Democrats.

Meanwhile, anti-tax sentiment is so strong in the House that one GOP freshman, Rep. Andy Harris (Md.), said he would vote against any tax increase that wasn’t paired with spending cuts at least 10 times as large. And if Obama rejects such a deal? “Then we go over the cliff,” Harris said.

Obama said Thursday that the refusal by House members to contemplate higher taxes for the rich remains the biggest impediment to an agreement. Obama won reelection in November after a campaign focused heavily on the issue, and polls show the public overwhelmingly supports his position.

“The big problem right now is that the Republicans in the House are resistant to the idea of the wealthiest Americans paying higher tax rates, and I understand they have a philosophical objection,” Obama told WCCO-TV, a CBS affiliate in Minneapolis-St. Paul.

Boehner’s meeting with Obama at the White House capped a week marked by increasingly testy relations between the two men, including a “tense” telephone conversation Tuesday evening. Though the pair met Sunday and appeared to make progress, optimism quickly evaporated as Republicans accused the White House of dropping its demand for new taxes from $1.6 trillion over the next decade to $1.2 trillion — and then jacking it back up to $1.4 trillion.

Boehner responded by sending the White House a “counter-offer” that changed little from his original offer calling for $800 billion in new taxes over the next decade. By all accounts, the two sides also remained far apart on spending cuts, with Obama offering about $600 billion over 10 years and Boehner demanding about $1.3 trillion.

The two men are trying to forge consensus on a debt-reduction plan that would replace more than $500 billion in automatic tax hikes and federal spending cuts set to take effect in January. Unless Congress acts, taxes will rise substantially for nearly 90 percent of Americans in 2013, an outcome that many economists say would throw the nation back into recession.

To many observers outside Washington, particularly on Wall Street, the dueling news conferences and public recriminations look suspiciously like pro forma posturing in advance of an eleventh-hour deal, a dance that has become common in Washington.

But negotiations over the fiscal cliff have been anything but usual. Obama and Boehner are largely working alone, and they have never successfully negotiated a major compromise without the help of other key players.

Most notable is the absence of Vice President Biden, who in recent days appears to have spent more time holding a photo op at the new Costco in Northeast Washington than working with his old colleagues in the Senate. Two years ago, Biden and McConnell negotiated a pact that extended tax cuts enacted during the George W. Bush administration until the end of this month. The two men reprised that performance in the summer of 2011, working with Senate Majority Leader Harry M. Reid (D-Nev.) to end an epic battle over the limit on federal borrowing.

Obama has been keeping Reid informed about the negotiations. The two men spoke again by phone Thursday. But there have been no late-night meetings, no weekend bargaining sessions.

Late Thursday, as Boehner and McConnell prepared to head to their home states for the weekend, the halls of the Capitol were virtually deserted.